arXiv subjects
Publications and source records attributed to Giovanni Falcone.
In this paper, we provide a complete classification of real three-dimensional simple Bol algebras $(B, [.,.], \langle .,.,. \rangle )$. The main part of the classification concerns the case $[B,B]=B$. In addition, we identify two isomorphism classes with a nonzero binary product, both satisfying $\dim[B,B]=2$, as well as four simple classes of Lie triple systems in which the binary product vanishes.
Thompson proved that every finite group admitting a fixed-point-free automorphism of prime order is nilpotent, and Kegel showed that the same conclusion holds for finite groups admitting a splitting automorphism of prime order. Motivated by these results, Sozutov asked whether a \(p'\)-group admitting a splitting automorphism of prime order is locally nilpotent if \[ \langle g, g^\varphi, \dots, g^{\varphi^{p-1}} \rangle \] is nilpotent for every \(g \in G\), \cite[Problem 10.59]{kourovka21}. We prove that if \(G\) is a periodic residually finite group admitting a splitting automorphism of prime order \(p\) then \(G\) is nilpotent of class bounded in terms of \(p\). This gives an affirmative answer, for residually finite groups, to the problem of Sozutov. We also prove that a possible counterexample to Sozutov's problem cannot be a Tarski monster.
The adaptation to users' preferences and the ability to infer and interpret humans' beliefs and intents, which is known as the Theory of Mind (ToM), are two crucial aspects for achieving effective human-robot collaboration. Despite its importance, very few studies have investigated the impact of adaptive robots with ToM abilities. In this work, we present an exploratory comparative study to investigate how social robots equipped with ToM abilities impact users' performance and perception. We design a two-layer architecture. The Q-learning agent on the first layer learns the robot's higher-level behaviour. On the second layer, a heuristic-based ToM infers the user's intended strategy and is responsible for implementing the robot's assistance, as well as providing the motivation behind its choice. We conducted a user study in a real-world setting, involving 56 participants who interacted with either an adaptive robot capable of ToM, or with a robot lacking such abilities. Our findings suggest that participants in the ToM condition performed better, accepted the robot's assistance more often, and perceived its ability to adapt, predict and recognise their intents to a higher degree. Our preliminary insights could inform future research and pave the way for designing more complex computation architectures for adaptive behaviour with ToM capabilities.
In this article we study extensions of Steiner triple systems by means of the associated Steiner loops. We recognize that the set of Veblen points of a Steiner triple system corresponds to the center of the Steiner loop. We investigate extensions of Steiner loops, focusing in particular on the case of Schreier extensions, which provide a powerful method for constructing Steiner triple systems containing Veblen points.
For an (imaginary) hyperelliptic curve $ \mathcal{H} $ of genus $g$, with a Weierstrass point $Ω$, taken as the point at infinity, we determine a basis of the Riemann-Roch space $\mathcal{L}(Δ+ m Ω)$, where $Δ$ is of degree zero, directly from the Mumford representation of $Δ$. This provides in turn a generating matrix of a Goppa code.
For an (imaginary) hyperelliptic curve $\mathcal{H}$ of genus $g$, we determine a basis of the Riemann-Roch space $\mathcal{L}(D)$, where $D$ is a divisor with positive degree $n$, linearly equivalent to $P_1+\cdots+ P_j+(n-j)Ω$, with $0 \le j \le g$, where $Ω$ is a Weierstrass point, taken as the point at infinity. As an application, we determine a generator matrix of a Goppa code for $j=g=3$ and $n=4.$
We study finite size scaling for the magnetic observables of an impurity residing at the endpoint of an open quantum Ising chain in a transverse magnetic field, realized by locally rescaling the magnetic field by a factor $μ\neq 1$. In the homogeneous chain limit at $μ= 1$, we find the expected finite size scaling for the longitudinal impurity magnetization, with no specific scaling for the transverse magnetization. At variance, in the classical impurity limit, $μ= 0$, we recover finite scaling for the longitudinal magnetization, while the transverse one basically does not scale. For this case, we provide both analytic approximate expressions for the magnetization and the susceptibility as well as numerical evidences for the scaling behavior. At intermediate values of $μ$, finite size scaling is violated, and we provide a possible explanation of this result in terms of the appearance of a second, impurity related length scale. Finally, on going along the standard quantum-to-classical mapping between statistical models, we derive the classical counterpart of the quantum Ising chain with an impurity at its endpoint as a classical Ising model on a square lattice wrapped on a half-infinite cylinder, with the links along the first circle modified as a function of $μ$.
We classify finite-dimensional nilpotent Lie algebras with $2$-dimensional central commutator ideals admitting a Lie group of automorphisms isomorphic to $SO_2(\mathbb R)$. This enables one to enlarge the class of nilpotent Lie algebras of type {n,2}.
We determine the algebraic structure of the multiplicative loops for locally compact $2$-dimensional topological connected quasifields. In particular, our attention turns to multiplicative loops which have either a normal subloop of positive dimension or which contain a $1$-dimensional compact subgroup. In the last section we determine explicitly the quasifields which coordinatize locally compact translation planes of dimension $4$ admitting an at least $7$-dimensional Lie group as collineation group.
We try to embed a t-design in a finite commutative group in such a way that the sum of the k points of a block is zero. We can compute the number of blocks of the boolean 2-design having all the non zero vectors of $(Z_2)^n$ as the set of points and the k-subsets of elements the sum of which is zero as blocks.
The bare-dressed technique is, for the first time, used in collision theory. The approach is valid for classical as well for quantum binary elastic collisions in the non relativistic regime. The same formalism can be used for inelastic collisions, when the particles undergo only a change of their internal quantum states during the collision process. All kinematic results can be obtained by a simple matrix transformation. Moreover, to make also simple and clear the results, for the inelastic collisions, the restitution coefficient formulation of inelastic processes has been used.
We give a sufficient and necessary condition for a p-adic integer to have p-th root in the ring of p-adic integers. The same condition holds clearly for residues modulo p^k. We give a proof that Fermat's last theorem is false for p-adic integers and for residues mod p^k.
Given two cyclotomic polynomials $Φ_n(x)$ and $Φ_m(x)$, $n\not= m$, we determine the minimal natural number k such that we can write $$k=a(x)Φ_n (x)+b(x)Φ_m(x),$$ with a(x) and b(x) integer polynomials.
We have applied a double decoupled localized level Anderson-Newns Hamiltonian to the analysis of surface effects upon the ionized fraction $\mathcal{R}_{+}$ of sputtered atoms from a metal surface. Electronic excitations, induced in the conduction band by the transient formation of quasi molecular systems, between substrate and emitted atoms, in the collision cascade generated by the primary incident beam, have been explicitly included into an instantaneous transition matrix peaked at the Fermi level of the material. The interaction dynamics seem to take place over two different time scales, one related to sputtered atom trajectories and the other to recoiled substrate particles. Finite temperature calculations have suggested, at very low ejection energies, a power law dependence of the final charge state of the sputtered beam on its detected velocity. This result is in agreement, in the zero temperature limit, with some previously published papers and its validity has been compared to other theoretical outcomes and tested on SIMS data.